Every prime ideal is max. if anysequence $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots $ stabilises?

Question

Let $R$ be a comutative ring with $1 \neq 0 $. Assume that for every sequence of ideals $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots $ there is a $N$ such that $$ n,m \geq N \quad \Longrightarrow \quad I_n=I_m $$ Show that every prime ideal in this ring is maximal.

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I don't know what I can do to prove it. A small hint please?

Answer 1

Your ring is called Artinian.

Let $\mathfrak{p}$ be a prime ideal of $R$, then $R/\mathfrak{p}$ is an Artinian integral domain. For any $x$, consider the sequence of principal ideals of $R/\mathfrak{p}$ $$(x)\supseteq (x^2)\supseteq\cdots\supseteq (x^n)\supseteq\cdots$$ The fact that sequence stops implies ...